www.downloadslide.com 606 Chapter 11 Linear Models and Estimation by Least Squares We want a confidence interval when x = 1.5; therefore, x ∗ = 1.5 and ˆβ 0 + ˆβ 1 x ∗ = 2.563 − (1.056)(1.5) = .979. Using calculations from parts (a) and (b), we obtain the desired 90% confidence interval: √ 1 (1.5 − 1.457)2 .979 ± (2.132)(.045) + , or (.938, 1.020). 6 .234 Thus, we would estimate that the mean strength of concrete with a water/cement ratio of 1.5 to be between .938 and 1.020. We can see from the variance expression that the confidence interval gets wider as x ∗ gets farther from x = 1.457. Also, the values x ∗ = .3 and x ∗ = 2.7 are far from the values that were used in the experiment. Considerable caution should be used before constructing a confidence interval for E(Y ) when the values of x ∗ are far removed from the experimental region. Water/cement ratios of .3 and 2.7 would probably yield concrete that is utterly useless! In many real-world situations, the most appropriate deterministic component of a model is not linear. For example, many populations of plants or animals tend to grow at exponential rates. If Y t denotes the size of the population at time t, we might employ the model E(Y t ) = α 0 e α 1t . Although this expression is not linear in the parameters α 0 and α 1 , it can be linearized by taking natural logarithms. If Y t can be observed for various values of t, wecan write the model as ln Y t = ln α 0 + α 1 t + ε and estimate ln α 0 and α 1 by the method of least squares. Other basic models can also be linearized. In the biological sciences, it is sometimes possible to relate the weight (or volume) of an organism to some linear measurement such as length (or weight). If W denotes weight and l length, the model E(W ) = α 0 l α 1 for unknown α 0 and α 1 is often applicable. (This model is known as an allometric equation.) If we want to relate the weight of randomly selected organisms to observable fixed lengths, we can take logarithms and obtain the linear model ln W = ln α 0 + α 1 ln l + ε = β 0 + β 1 x + ε with x = ln l. Then, β 0 = ln α 0 and β 1 = α 1 can be estimated by the method of least squares. The following example illustrates such a model. EXAMPLE 11.11 In the data set of Table 11.5, W denotes the weight (in pounds) and l the length (in inches) for 15 alligators captured in central Florida. Because l is easier to observe (perhaps from a photograph) than W for alligators in their natural habitat, we want to